The Gamma and Chi-Square Distributions

STA 211 · Lecture 15

Introduction

The gamma family models positive, right-skewed quantities such as waiting times, lifetimes and aggregate durations. The exponential distribution is a gamma special case, while the chi-square distribution is a re-parameterised gamma distribution central to inference about variances.

Learning Objectives

You should be able to:

• define and use the gamma function;
• state the gamma density under a clear parameterisation;
• calculate gamma probabilities, means, variances and MGFs;
• explain exponential and chi-square special cases; and
• use additivity within the gamma family.

Gamma Function

For $\alpha > 0$,

$$\Gamma(\alpha) = \int_{0}^{\infty}x^{\alpha - 1}e^{- x}\, dx.$$

The recurrence relation is

$$\Gamma(\alpha + 1) = \alpha\Gamma(\alpha).$$

For positive integers $n$,

$$\Gamma(n) = (n - 1)!.$$

Also,

$$\Gamma\left( \frac{1}{2} \right) = \sqrt{\pi}.$$

Gamma Distribution: Shape-Rate Form

This course uses the shape-rate parameterisation. If $X \sim Gamma(\alpha,\beta)$, where $\alpha > 0$ is the shape and $\beta > 0$ is the rate, then

$$f_{X}(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha - 1}e^{- \beta x},\quad\quad x > 0.$$

Gamma probability density functions

Gamma probability density functions

The mean and variance are

$$E(X) = \frac{\alpha}{\beta},$$

$$Var(X) = \frac{\alpha}{\beta^{2}}.$$

The MGF is

$$M_{X}(t) = \left( \frac{\beta}{\beta - t} \right)^{\alpha},\quad\quad t < \beta.$$

Shape-scale notation

Some texts use scale $\theta = 1/\beta$. Then

$$f_{X}(x) = \frac{x^{\alpha - 1}e^{- x/\theta}}{\Gamma(\alpha)\theta^{\alpha}},$$

$$E(X) = \alpha\theta,\quad\quad Var(X) = \alpha\theta^{2}.$$

The two parameterisations must not be mixed.

Exponential Distribution

When $\alpha = 1$,

$$f_{X}(x) = \beta e^{- \beta x},\quad\quad x > 0,$$

which is the exponential distribution with rate $\beta$. Its survival function is

$$P(X > x) = e^{- \beta x}.$$

Like the geometric distribution, the exponential distribution is memoryless:

$$P(X > s + t \mid X > s) = P(X > t).$$

Additivity

If $X_{i} \sim Gamma\left( \alpha_{i},\beta \right)$ are independent with a common rate $\beta$, then

$$\sum_{i = 1}^{n}X_{i} \sim Gamma\left( \sum_{i = 1}^{n}\alpha_{i},\beta \right).$$

A common rate is required.

Example 15.1

If independent service stages have distributions $Gamma(2,0.5)$ and $Gamma(3,0.5)$, their total time has distribution

$$Gamma(5,0.5).$$

Its mean is $5/0.5 = 10$ and variance is $5/(0.5)^{2} = 20$.

Chi-Square Distribution

A chi-square variable with $\nu$ degrees of freedom is a gamma variable with

$$\alpha = \frac{\nu}{2},\quad\quad\beta = \frac{1}{2}.$$

Thus,

$$X \sim \chi_{\nu}^{2}\quad \Leftrightarrow \quad X \sim Gamma\left( \frac{\nu}{2},\frac{1}{2} \right).$$

Its density is

$$f_{X}(x) = \frac{1}{2^{\nu/2}\Gamma(\nu/2)}x^{\nu/2 - 1}e^{- x/2},\quad\quad x > 0.$$

Chi-square probability density functions

Chi-square probability density functions

The mean and variance are

$$E(X) = \nu,$$

$$Var(X) = 2\nu.$$

If $Z_{1},\ldots,Z_{\nu}$ are independent standard normal variables, then

$$\sum_{i = 1}^{\nu}Z_{i}^{2} \sim \chi_{\nu}^{2}.$$

If $X_{1},\ldots,X_{n}$ are sampled independently from $N\left( \mu,\sigma^{2} \right)$, then

$$\frac{(n - 1)S^{2}}{\sigma^{2}} \sim \chi_{n - 1}^{2}.$$

This result underlies confidence intervals and hypothesis tests for a normal population variance.

Post-Test

1. State the gamma density under the shape-rate parameterisation.
2. If $X \sim Gamma(4,2)$, find its mean and variance.
3. Identify the exponential distribution as a gamma special case.
4. If $X \sim \chi_{10}^{2}$, find its mean and variance.
5. If $X \sim Gamma(3,1/2)$, express its distribution as chi-square.
6. State the condition required for sums of independent gamma variables to remain gamma.

Selected answers: 2. $2$ and $1$; 4. $10$ and $20$; 5. $\chi_{6}^{2}$; 6. a common rate, or equivalently a common scale.

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